1402 Mystery Level

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Mystery level puzzles may be very difficult or relatively easy. How much trouble will this one be? You’ll have to try it to see!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you a little bit about the number 1402:

  • 1402 is a composite number.
  • Prime factorization: 1402 = 2 × 701.
  • 1402 has no exponents greater than 1 in its prime factorization, so √1402 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1402 has exactly 4 factors.
  • The factors of 1402 are outlined with their factor pair partners in the graphic below.

1402 is the hypotenuse of a Pythagorean triple:
31² + 21² = 1402

1402 is the hypotenuse of a Pythagorean triple:
520-1302-1402 which is 2 times (260-651-701)
and can also be calculated from 2(31)(21), 31² – 21², 31² + 21²

1401 Roasting Over an Open Fire

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I went camping last week. My family roasted hotdogs. Some people refer to them as mystery meat. Others roasted marshmallows. I was surprised to learn that almost all brands of marshmallows have blue dye in them.  I’m told that without that blue dye the marshmallows will lose their whiteness as they sit on store shelves. Why they have to be that white is a mystery to me.

Here’s a mystery level puzzle for you to solve. It looks a lot like the utensil that was used to roast the hotdogs and marshmallows.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll tell you something about the number 1401:

  • 1401 is a composite number.
  • Prime factorization: 1401 = 3 × 467.
  • 1401 has no exponents greater than 1 in its prime factorization, so √1401 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1401 has exactly 4 factors.
  • The factors of 1401 are outlined with their factor pair partners in the graphic below.

1401 is the difference of two squares in two different ways. Can you figure out what those ways are?

Prime Factorization of the Hundred Numbers up to 1400

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Almost one-third of the numbers from 1301 to 1400 have 4 factors. Only 1/5 of the numbers have 8 factors.

Since 1/3 is significantly bigger than 1/5, the amount of factors for these numbers wouldn’t make a very exciting horse race. Here is the breakdown:

  • 11 numbers had 2 factors
  • 1 number had 3 factors
  • 32 numbers had 4 factors
  • 7 numbers had 6 factors
  • 20 numbers had 8 factors
  • 2 numbers had 10 factors
  • 13 numbers had 12 factors
  • 4 numbers had 16 factors
  • 1 number had 18 factors
  • 2 numbers had 20 factors
  • 5 numbers had 24 factors
  • 1 number had 28 factors
  • 1 number had 32 factors

The rosy looking numbers have square roots that can be simplified, and that is only 37% of the numbers listed.

You may not expect it, but 1400 is one of the numbers with 24 factors. Let me tell you a little bit about 1400 and why it has so many factors:

  • 1400 is a composite number.
  • Prime factorization: 1400 = 2 × 2 × 2 × 5 × 5 × 7, which can be written 1400 = 2³ × 5² × 7
  • 1400 has at least one exponent greater than 1 in its prime factorization so √1400 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1400 = (√100)(√14) = 10√14
  • The exponents in the prime factorization are 3, 2, and 1. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1400 has exactly 24 factors.
  • The factors of 1400 are outlined with their factor pair partners in the graphic below.

1400 is the hypotenuse of TWO Pythagorean triples:
392-1344-1400 which is (7-24-25) times 56
840-1120-1400 which is (3-4-5) times 280

1399 and Level 6

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The eligible common factors of 48 and 72 are 6, 8, and 12. The common factors for 10 and 30 are 5 and 10.  Don’t guess and check the possibilities! Can you figure out the logic needed to start this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s a little information about the number 1399:

  • 1399 is a prime number.
  • Prime factorization: 1399 is prime.
  • 1399 has no exponents greater than 1 in its prime factorization, so √1399 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1399 has exactly 2 factors.
  • The factors of 1399 are outlined with their factor pair partners in the graphic below.

How do we know that 1399 is a prime number? If 1399 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1399. Since 1399 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1399 is a prime number.

1399 is the difference of two squares:
700² – 699² = 1399

 

1398 and Level 5

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You might find this puzzle to be a little tricky, but if you always use logic before you write any of the factors, you should succeed!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here is some information about the number 1398:

  • 1398 is a composite number.
  • Prime factorization: 1398 = 2 × 3 × 233
  • 1398 has no exponents greater than 1 in its prime factorization, so √1398 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1398 has exactly 8 factors.
  • The factors of 1398 are outlined with their factor pair partners in the graphic below.

1398 is the hypotenuse of a Pythagorean triple:
630-1248-1398 which is 6 times (105-208-233)

1397 and Level 4

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I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1397:

  • 1397 is a composite number.
  • Prime factorization: 1397 = 11 × 127
  • 1397 has no exponents greater than 1 in its prime factorization, so √1397 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1397 has exactly 4 factors.
  • The factors of 1397 are outlined with their factor pair partners in the graphic below.

1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397

31 Flavors of 1396

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The first 52 triangular numbers are 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378.

OEIS.org informs us that 1396 can be written as the sum of three triangular numbers in 31 different ways. It is the smallest number that can make that claim!

That 31st way is written with three consecutive triangular numbers, 435, 465, and 496, which are the 29th, 30th, and 31st triangular numbers respectively. That fact makes 1396 the 31st Centered Triangular Number as well!

That is, at least, 1396 is the 31st number on the list. You can also calculate it using this formula: [3(30²) + 3(30) + 2]/2 = 1396

Here’s more about the number 1396:

  • 1396 is a composite number.
  • Prime factorization: 1396 = 2 × 2 × 349, which can be written 1396 = 2² × 349
  • 1396 has at least one exponent greater than 1 in its prime factorization so √1396 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1396 = (√4)(√349) = 2√349
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1396 has exactly 6 factors.
  • The factors of 1396 are outlined with their factor pair partners in the graphic below.

1396 is the sum of two squares:
36² + 10² = 1396

1396 is the hypotenuse of a Pythagorean triple:
720-1196-1396 calculated from 2(36)(10), 36² – 10², 36² + 10²

1395 and Level 2

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1391 is the 22nd Friedman number, and there are TWO reasons why!

See! Factoring numbers can be such an exciting adventure! Can you find the factors for this puzzle?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403\

Here’s more about the number 1395:

  • 1395 is a composite number.
  • Prime factorization: 1395 = 3 × 3 × 5 × 31, which can be written 1395 = 3² × 5 × 31
  • 1395 has at least one exponent greater than 1 in its prime factorization so √1395 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1395 = (√9)(√155) = 3√155
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1395 has exactly 12 factors.
  • The factors of 1395 are outlined with their factor pair partners in the graphic below.

You can see the reasons 1395 is the 22nd Friedman numbers in these factor pairs:
15 × 93 = 1395
45 × 31 = 5×9×31 = 1395, that one uses the digits in reverse order!

1395 is also the hypotenuse of a Pythagorean triple:
837-1116-1395 which is (3-4-5) times 279

 

1394 and Level 2

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The factors and most of the products are missing from this multiplication table, and the ones that are there aren’t in there usual places. Can you figure out where everything goes?

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Now I’ll write a little bit about the puzzle number, 1394:

  • 1394 is a composite number.
  • Prime factorization: 1394 = 2 × 17 × 41
  • 1394 has no exponents greater than 1 in its prime factorization, so √1394 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1394 has exactly 8 factors.
  • The factors of 1394 are outlined with their factor pair partners in the graphic below.

1394 is the hypotenuse of FOUR Pythagorean triples:
306-1360-1394 which is (9-40-41) times 34
370-1344-1394 which is 2 times (185-672-697)
656-1230-1394 which is (8-15-17) times 82
910-1056-1394 which is 2 times (455-528-697)

1393 DNA Shared Matches

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Ancestry.com gave my husband a list of his 50 top matches of DNA from their database. For each match they found, I could click on a button that would reveal any matches that my husband shared with that match. Some of his matches didn’t share any other match with him. Sometimes a couple of their shared matches didn’t make his list of top 50 matches. I made a table of his shared matches. It was pretty big so I made a smaller table that only includes people in his top 50 who have at least one shared match with him AND a second or third cousin.

I purposely cut off people’s names for privacy reasons, but anyone who shares DNA with my husband and the others in the table should still be able to figure out who’s who.

Ancestry explains that a 2nd cousin could actually be a great aunt or a 1st cousin twice removed. The 2nd cousin would have 5 to 6 degrees of separation from my husband, a 3rd cousin would have 6 to 10 degrees of separation, and a 4th cousin would have 6 to 12 degrees of separation, but most likely 10.

DNA does NOT “share and share alike”. Every person gets half of his DNA from his mother and a half from his father, but the half given from each parent can vary from child to child. I noticed that some of my husband’s matches might be siblings with the same surname, but their shared matches were not always the same. Thus, it can definitely be worth it to have more than one family member take the DNA test.

I made this chart to see if it could help me determine who might be my husband’s maternal cousins versus his paternal cousins. I don’t think I completely succeeded. The same DNA might not be the DNA in shared matches. For example, ab, bc, and ac each share a letter of the alphabet with each other, but it is not the same letter of the alphabet. Since both sides of my husband’s family had many siblings and cousins and settled in the Cleveland, Ohio area 100 years ago or more, it seems possible that some of his relatives listed on the chart are actually related to BOTH his father and his mother, but more distantly than 4th cousin on either side.

A positive from making the chart is that I have verified that all the people with x’s in the lower right corner are closely related to each other. The chart says they are also all related to Benjam, but none of them have any idea how.

Like so much of genealogy research, one answer will produce more questions. It becomes such a fascinating puzzle!

Since this is my 1393rd post, I’ll write a little bit about that number:

  • 1393 is a composite number.
  • Prime factorization: 1393 = 7 × 199
  • 1393 has no exponents greater than 1 in its prime factorization, so √1393 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1393 has exactly 4 factors.
  • The factors of 1393 are outlined with their factor pair partners in the graphic below.

Since both of its factor pairs have odd numbers in it, I know that 1393 can be written as the difference of two squares in two ways:
697² – 696² = 1393
103² – 96² = 1393